Space-filling tree

Space-filling trees are geometric constructions that are analogous to space-filling curves, but have a branching, tree-like structure and are rooted. A space-filling tree is defined by an incremental process that results in a tree for which every point in the space has a finite-length path that converges to it. In contrast to space-filling curves, individual paths in the tree are short, allowing any part of the space to be reached from the root; the simplest examples of space-filling trees have a regular, self-similar, fractal structure, but can be generalized to non-regular and randomized/Monte-Carlo variants. Space-filling trees have interesting parallels in nature, including fluid distribution systems, vascular networks, fractal plant growth, many interesting connections to L-systems in computer science. A space-filling tree is defined by an iterative process whereby a single point in a continuous space is connected via a continuous path to every other point in the space by a path of finite length, for every point in the space, there is at least one path that converges to it.

The term "space-filling tree" in this sense was created in a 2009 tech report that defines "space-filling" and "tree" differently than their traditional definitions in mathematics. As explained in the space-filling curve article, in 1890, Peano found the first space-filling curve, by Jordan's 1887 definition, now standard, a curve is a single function, not a sequence of functions; the curve is "space filling" because it is "a curve whose range contains the entire 2-dimensional unit square". In contrast, a space-filling tree, as defined in the tech report, is not a single tree, it is only a sequence of trees. The paper says "A space-filling tree is defined as an infinite sequence of trees", it defines T square as a "sequence of trees" states " T square is a space-filling tree". It is not space-filling in the standard sense of including the entire 2-dimensional unit square. Instead, the paper defines it as having trees in the sequence coming arbitrarily close to every point, it states "A tree sequence T is called'space filling' in a space X if for every x ∈ X, there exists a path in the tree that starts at the root and converges to x.".

The standard term for this concept is that it includes a set of points, dense everywhere in the unit square. The simplest example of a space-filling tree is one; the images illustrate the construction for the planar region 2 ⊂ R 2. At each iteration, additional branches are added to the existing trees. Space-filling trees can be defined for a variety of other shapes and volumes. Below is the subdivision scheme used to define a space-filling for a triangular region. At each iteration, additional branches are added to the existing trees connecting the center of each triangle to the centers of the four subtriangles; the first six iterations of the triangle space-filling tree are illustrated below: Space-filling trees can be constructed in higher dimensions. The simplest examples are hypercubes in R n. A similar sequence of iterations used for the square space-filling tree can be used for hypercubes; the third iteration of such a space-filling tree in R 3 is illustrated below: H tree Space-filling curve Rapidly exploring random tree Binary space partitioning

Heart and Soul (Kathy Troccoli album)

Heart and Soul is the second studio album by singer-songwriter Kathy Troccoli. This album received a Grammy nomination in 1984, contains songs written by Michael W. Smith, it featured music popular with CCM, as well as synthpop friendly songs, "I Belong To You", "Holy Holy", "Hearts Of Fire". The album features the songwriting skills of Brown and Debbie Bannister, Amy Grant, Gary Chapman, Michael W. Smith, as well as others; the music is a blend of contemporary pop and gospel music. It was released as a vinyl LP record. "Long Distance Letter" - 3:13 "Yes And Know" - 3:55 "Bittersweetness" - 3:54 "Mighty Lord" - 5:02 "Open My Eyes" - 4:27 "Holy, Holy" - 5:01 "Hearts of Fire" - 4:00 "I Belong To You" - 4:16 "Island of Love" - 3:54 Producer – Brown Bannister Executive Producers – Dan Harrell and Michael Blanton Engineered and Mixed by Jack Joseph Puig Assistant Engineers – Jim Baird, Don Cobb, Steve Ford and Mike Psanos. Mastered by Doug Sax at The Mastering Lab. Kathy Troccoli – lead vocals Shane Keistersynthesizers, acoustic piano Keith Thomas – synthesizers, keyboards Michael W. Smith – synthesizers, acoustic piano Phil Madeira – synthesizers Dann Huff – guitars Mike Brignardellobass guitar Paul Leimdrums Lenny Castropercussion Ernie Wattssaxophone Terry McMillanharmonica Kathy Burdick – backing vocals Thom Flora – backing vocals Gary Pigg – backing vocals Kim Fleming – backing vocals Diana Hanna – backing vocals Donna McElroy – backing vocals Chris Harris – backing vocals Doug Clements – backing vocals Jackie Cusic – backing vocals Marty McCall – backing vocals Judy Rodman – backing vocals Melodie Tunney – backing vocals Kathy Troccoli, "Heart and Soul" liner notes.

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